The effect of flow on resonant absorption of slow MHD waves in magnetic flux tubes
aa r X i v : . [ phy s i c s . p l a s m - ph ] J a n The effect of flow on resonant absorption of slowMHD waves in magnetic flux tubes
Mohammad Sadeghi , Karam Bahari , Kayoomars Karami Department of Physics, University of Kurdistan, Pasdaran Street, P.O. Box 66177-15175, Sanandaj, Iran Physics Department, Faculty of Science, Razi University, Kermanshah, Iran
January 7, 2021
Abstract
In this paper, we study kink and sausage oscillations in the presence of longitudinalbackground flow. We study resonant absorption of the kink and sausage modes in theslow continuum under magnetic pore conditions in the presence of flow. we determine thedispersion relation then solve it numerically, and find the frequencies and damping rates ofthe slow kink and sausage surface modes. We also, obtain analytical solution for the dampingrate of the slow surface mode in the long wavelength limit. We show that in the presence ofplasma flow, resonance absorption can result in strong damping for forward waves and can beconsidered as an efficient mechanism to justify the extremely rapid damping of slow surfacesausage waves observed in magnetic pores. Also, the plasma flow reduces the efficiency ofresonance absorption to damp backward waves. Furthermore, for the pore conditions, theresonance instability is avoided in our model.
The mechanism of the heating of the solar corona (and the corona of the stars) is not yet fullyunderstood. Several non-thermal mechanisms have been proposed to explain this phenomenon,and the problem of justifying this phenomenon remains. Surely the heating must be tied to themagnetic field, because it is obvious that the heated areas have a non-potential magnetic field.Plasma is bounded by magnetic field lines and can form many types of visible structures. One ofthese is the propagation of magnetohydrodynamic (MHD) waves and their damping. Resonantabsorption proposed as the damping mechanism of MHD waves for the first time by Ionson [1].With the launch of space satellites, the interest of theoretical physicists in studying waves inthe solar atmosphere, and especially the use of resonance absorption, increased. Nakariakov re-ported transverse oscillations in coronal loops with high damping rate [2] . Ruderman & Robertsexpressed the idea that the observed period of oscillation and their damping time can be usedto determine the transverse density distribution in a coronal magnetic loop [3]. This methodwas later used by many researchers (e.g., [4]- [17]).Because the source of the high-temperature energy of the corona originates from the convec-tion zone below the surface of the sun, it is important to study the dynamics of MHD waves inthe photosphere and chromosphere (e.g., [18]; [19]). In the photosphere, in addition to Alfv´enresonance, energy transfer by slow resonance absorption can be of particular importance. Yu etal. showed that slow resonance absorption can affect the damping of waves in the photosphere[21] . They also found that the resonant damping of the fast surface kink mode is much stronger1han that of the slow surface kink mode. Yu et al. [20] considered linear profile for densityand pressure in the transitional layers [20]. They showed in the cases where damping by Alfv´encontinuum is weak, the resonant absorption in slow continuum can be an effective mechanismfor damping sausage and kink slow surface modes. Sadeghi & Karami investigated resonanceabsorption in the presence of a weak magnetic twist in the photosphere condition [22]. Theyconcluded that a magnetic twist could be effective on more intense damping. In this paper, westudy effect of flow on the slow sausage and kink MHD waves, which have been observed byDunn Solar Telescope [23].Observations by Brekke et al. and Tian et al. show that plasma flows in magnetic flux tubesare present everywhere in the solar atmosphere [24] and [31]. Soler et al. reported that the flowvelocities are usually less than 10% of the plasma Alfv´en speed [32]. Grant et al. investigatedwave damping observed in upwardly propagating sausage mode oscillations contained within amagnetic pore [23]. They showed that the waves propagate only through 0.25 of it’s wavelengthalong the before they damp whereas theory would expect the wave to survive for the distanceof a few wavelengths. They also showed that the average upflow speed in photosphere is about1 / Equations of Motion and Model
The linear perturbations of homogeneous flowing magnetized plasma are governed by the fol-lowing equations [40] (1a) ρ (cid:18) ∂∂t + v · ∇ (cid:19) ξ = −∇ δp − µ (cid:16) δ B × ( ∇ × B ) + B × ( ∇ × δ B ) (cid:17) , (1b) δp = − ξ · ∇ p − γp ∇ · ξ , (1c) δ B = −∇ × ( B × ξ ) , where ρ, p , v and B are the background density, kinetic pressure, plasma velocity and magneticfield, respectively. Also ξ is the Lagrangian displacement vector, δp and δ B are the Eulerianperturbations of the pressure and magnetic field, respectively. Here, γ is the ratio of specificheats (taken to be 5 / µ is the permeability of free space.We consider a flux tube model with a unidirectional magnetic field which is in the directionof the tube axis. The model consists of interior and exterior regions in which the equilibriumand stationary quantities are constant and transitional layer in which the background quantitiesvary continuously. In the cylindrical coordinate the magnetic field is (2) B = (cid:16) , , B z ( r ) (cid:17) . Plasma pressure and magnetic field must be satisfied in the hydrostatic equilibrium equation(3)dd r (cid:18) p + B z µ (cid:19) = 0 . Here the background plasma density and magnetic field are assumed to be the same as thoseconsidered by Sadeghi & Karami (2019) [22] ρ ( r ) = ρ i , r r i ,ρ i + ( ρ e − ρ i ) (cid:16) r − r i r e − r i (cid:17) , r i < r < r e ,ρ e , r > r e , (4)where r i = R − l/ r e = R + l/
2. Here, R and l are the tube radius and the thickness ofthe inhomogeneous layer, respectively, B z ( r ) = B z i , r r i ,B z i + (cid:0) B z e − B z i (cid:1) (cid:16) r − r i r e − r i (cid:17) , r i < r < r e ,B z e , r > r e , (5)where ρ i and ρ e are the constant densities of the interior and exterior regions of the fluxtube, respectively. Also B z i and B z e are the interior and exterior constant longitudinal magneticfields, respectively. Putting Eqs. (5) into the magnetohydrostatic equation (3), we obtain thebackground gas pressure as follows p ( r ) = p i , r r i ,p i + ( p e − p i ) (cid:16) r − r i r e − r i (cid:17) , r i < r < r e ,p e , r > r e , (6)3here p e = p i + (cid:0) B z e − B z i (cid:1) µ , (7)and p i is an arbitrary constant. The plasma flow is considered to be in the direction of themagnetic field lines. as follows v z ( r ) = v z i , r r i ,v z i + ( v z e − v z i ) (cid:16) r − r i r e − r i (cid:17) , r i < r < r e ,v z e , r > r e , (8)where v zi and v ze are the constant flow of the interior and exterior regions of the flux tube,respectively. In addition, we define the following quantities v A ( i,e ) ≡ B z ( i,e ) µ ρ ( i,e ) , (9) v s ( i,e ) ≡ γ p ( i,e ) ρ ( i,e ) , (10) v c ( i,e ) ≡ v s ( i,e ) v A ( i,e ) v s ( i,e ) + v A ( i,e ) , (11)where v A ( i,e ) , v s ( i,e ) and v c ( i,e ) are the interior/exterior Alfv´en, sound and cusp velocities, re-spectively.Since the hydrostatic equilibrium is only a function of r, all the perturbed quantities including ξ and δP T can be Fourier analyzed (12)( ξ , δP T ) ∝ e i ( mφ + k z z − ωt ) , where ω is the oscillation frequency, m is the azimuthal wavenumber for which only integer valuesare allowed and, k z , is the longitudinal wavenumber in the z direction. We study both forwardand backward waves which propagate in the positive and negative z directions respectively,for both the waves the longitudinal wavenumber is restricted to positive values, the oscillationfrequency is positive for forward waves and is negative for backward wave. The perturbedquantity δP T = δp + B .δ B /µ is the Eulerian perturbation of total (gas and magnetic) pressure.Putting Eq. (12) into (1a)-(1c), we obtain the two coupled first order differential equations(13a) D d( r ξ )d r = − rC δP T , (13b) D d δP T d r = C ξ . The above equations derived earlier by Appert et al. [41] and later by Hain & Lust [42],Goedbloed [43] and Sakurai et al. [44]. Here, the multiplicative factors are defined as (14a)
D ≡ ρ (cid:0) v s + v A (cid:1) (cid:0) Ω − ω A (cid:1) (cid:0) Ω − ω A (cid:1) , (14b) C ≡ Ω − (cid:18) k z + m r (cid:19) (cid:0) v s + v A (cid:1) (cid:0) Ω − ω A (cid:1) , (14c) C ≡ ρ D (cid:0) Ω − ω A (cid:1) , in which 4 B ≡ k z B z ,ω A ≡ f B µ ρ . and ω c ≡ (cid:18) v s v A + v s (cid:19) ω A , Here Ω = ω − ω f is the Doppler shifted frequency which ω f (= k z v z . ) is the flow frequency, ω A (= k z v A ) is the Alfv´en oscillation frequency and ω c (= k z v c ) is the cusp oscillation frequency.Also v A = | B z | / √ µ ρ is the Alfv´en speed, v s = p γp/ρ is the sound speed, and v c = v s v A ( v s + v A ) / is the cusp speed.Combining Eqs. (13a) and (13b), one can obtain a second-order ordinary differential equationfor radial component of the differential equation for δP T as [45] (15)d δP T d r + 1 r d δP T d r − (cid:18) k r + m r (cid:19) δP T = 0 , where k r ≡ ( ω s − Ω )( ω A − Ω )( v A + v s )( ω c − Ω ) , (16)solutions of Eq. (15) in the interior ( r r i ) and exterior ( r > r e ) regions are given by (17a) δP T i ( r ) = A i I m ( k ri r ) , (17b) δP T e ( r ) = A e K m ( k re r ) , where A i and A e are constant. Also I ( . ) and K ( . ) are the modified Bessel function of the secondkind respectively. Replacing the solutions (17a) and (17b) into Eq. (13b) radial displacementcan be determined as (18a) ξ ri ( r ) = A i ρ i (Ω − ω Ai ) I ′ m ( k ri r ) , (18b) ξ re ( r ) = A e ρ i (Ω − ω Ae ) K ′ m ( k ri r ) , in which prime denotes differentiation of the function with respect to its argument. Thesesolutions are used in the next sections to determine the dispersion relation of the tube oscillations. In this section we consider a flux tube without the inhomogeneous layer and obtain the dispersionrelation of oscillations. For this purpose, the solutions obtained for ξ r and δP T in the last sectioninside and outside the tube (i.e Eqs. (17a)-(18b)) must be satisfied in the following boundaryconditions (19a) ξ ri (cid:12)(cid:12)(cid:12) r = R = ξ re (cid:12)(cid:12)(cid:12) r = R , (19b) δP T i (cid:12)(cid:12)(cid:12) r = R = δP T e (cid:12)(cid:12)(cid:12) r = R , where R is the tube radius. Then the dispersion relation can be determined after some algebraas ρ i (cid:0) Ω i − ω Ai (cid:1) − k ri k re ρ e (cid:0) Ω e − ω Ae (cid:1) Q m = 0 , (20)5 k z R ci / si / s i v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (a) k z R ci / si / s i v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (b) k z R - ci / si -0.862-0.86-0.858-0.856-0.854-0.852-0.85-0.848-0.846-0.844 / s i v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (c) k z R - ci / si -0.862-0.86-0.858-0.856-0.854-0.852-0.85-0.848-0.846-0.844 / s i v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (d) Figure 1: The Dopller shifted phase speed Ω /ω si , Eq. (20), of the slow surface sausage and kinkmodes versus k z R for various flow parameters v zi /v si for forward and backward waves. Panels (a)and (b) are for forward sausage and kink modes and panels (c) and (d) are for backward sausageand kink modes respectively. Under the magnetic pore conditions, following [23] the auxiliaryparameters are taken as v Ai = 12 km s − , v Ae = 0 km s − (i.e. B ze = 0), v si = 7 km s − , v se = 11 . − , v ci = 6 . − ( ≃ . v si ) and v ce = 0 km s − .6here Q m = I ′ m ( k ri R ) K m ( k re R ) I m ( k ri R ) K ′ m ( k re R ) . For the case with no flow (Ω i = Ω e = ω ), the dispersion relation reduces to the resultobtained by Edwin & Roberts [45] and Yu et al. [20].Here we solve the dispersion relation (20) numerically and the phase speed Ω /ω si of the slowsurface sausage ( m = 0) and kink ( m = 1) modes versus k z R for various values of the flowparameters v zi /v si are displayed in Fig. 1. Panels (a) and (b) are for forward sausage and kinkmodes and panels (c) and (d) are for backward sausage and kink modes respectively. The figureshows that (i) for a given value of k z R , for forward waves when the flow speed increases theDoppler shifted phase speed decreases and for backward waves the magnitude of the phase speedincreases. (ii) For a given flow speed v zi /v si as k z R increases the Doppler shifted phase speedfor forward decreases and magnitude of the Doppler shifted phase speed for backward increases.(iii) For k z R ≪
1, for both the forward and backward waves Ω /ω si tends to ω ci /ω si . (iv) Theseresults show that for specific values of the flow speed, the Doppler shifted phase speed is betweenthe internal and external values of the cusp speed of the flux tube. (vi) For the case of no flow,the result of Yu et al. [20] is recovered. In this section we consider a flux tube with an inhomogeneous boundary layer. According toEquations (4)-(6), the density, magnetic field and pressure change continuously from the insideto the outside of the tube, so in this case, the Dopller shifted (Ω) of the waves may be equal to thecusp ( ω c ) or Alfv´en ( ω A ) frequency. According to Yu et al. [20], under photosphere conditionsthe oscillation frequency will be equal to the cusp frequency at a point in the boundary layerwhich causes a singularity in the equations of motion. This phenomenon is called cusp resonantabsorption.Sakurai et al. [44] showed that under the thin boundary approximation, the solutions insideand outside the tube can be connected using the connection formula (21a)[ ξ r ] ≡ ξ re ( r e ) − ξ ri ( r i )= − iπ Sign Ω | ∆ c | µω c rB ω A (cid:12)(cid:12)(cid:12) r = r c δP T i , (21b)[ δP T ] ≡ δP T e ( r e ) − δP T i ( r i )= 0 , where [ ξ r ] and [ δP T ] represent the jumps for the Lagrangian radial displacement and total pres-sure perturbation across the inhomogeneous (resonant) boundary, which connects the solutionsinside and outside of the flux tube. The subscript c in ∆ c shows that the quantity must becalculated in the surface where the cusp resonance occurs. We will determine the location of thecusp resonance, r c later. We obtain the dispersion relation in the presence of flow by substitutingthe solutions (17a)-(18b) into the connection formula (21a) and (21b), the result is (22) ρ i (cid:0) Ω i − ω Ai (cid:1) − ρ e (cid:0) Ω e − ω Ae (cid:1) k i k e Q m + iπ Sign Ω | ∆ c | k z ρ (cid:18) v s v s + v A (cid:19) ρ i ρ e (cid:0) Ω i − ω Ai (cid:1) (cid:0) Ω e − ω Ae (cid:1) G m k e = 0 , v / v s i v ce v ci v si v se v Ai v s v A v c (a) v / v s i v Ai v se v si v ci v ce v s v A v c (b) Figure 2: Variations the sound v s ( − v s ), Alfv´en v A ( − v A ) and cusp speeds v c ( − v c ) versus δ in the annulus layer under magnetic pore conditions ( v Ai = 12 km s − , v Ae = 0 km s − , v ze = 0 km s − , v si = 7 km s − , v se = 11 . − , v ci = 6 . − ( ≃ . v si ) and v ce = 0 km s − ). When v ci ≤ v ≤ v cm resonance absorption occurs for the slow body modesand when v < v ci resonance absorption occurs for slow surface modes in the slow continuum.where G m = K m ( k re r e ) K ′ m ( k re r e ) . It is clear that in the absence of plasma flow this equation reduces tothe dispersion relation obtained by Yu et al. [20].To display the background quantities in the boundary layer we define the variable δ ≡ r − r i r e − r i which varies from 0 to 1 in the boundary layer. Using Eqs. (4) to (6), one can write thequantities v s = p γp/ρ , v A = | B z | / √ µ ρ in the inhomogeneous boundary layer as functions of δ as (23) v s = v si (cid:20) δ ( χv sei − δ ( χ − (cid:21) , (24) v A = v Ai (cid:20) δ ( χv Aei − δ ( χ − (cid:21) , and the cusp velocity v c ≡ v s v A ( v s + v A ) / in the inhomogeneous layer ( r i < r < r e ) as (25) v c = v si v Ai h δ ( χv sei − ih δ ( χv Aei − ih δ ( χ − ih v si (cid:16) δ ( χv sei − (cid:17) + v Ai (cid:16) δ ( χv Aei − (cid:17)i , where χ ≡ ρ e /ρ i , v sei ≡ v se /v si , v Aei ≡ v Ae /v Ai . Using Eqs. (23)-(25) we plot the sound,Alfv´en and cusp velocities under magnetic pore conditions in Fig. 2. The Figure shows thatfor v c < v ci and v ci < v c < v c max , the surface and body sausage modes can resonantly dampin the slow continuum respectively. Here, v c max is the maximum value of the cusp speed in thetransition layer.Note that according to Yu et al. [20], the position of the cusp resonance point r c is obtainedby setting Ω = ω c (cid:12)(cid:12)(cid:12) r = r c ≡ k z v c (cid:12)(cid:12)(cid:12) r = r c . Consequently, the resulting equation in terms of the8ariable δ c ≡ δ (cid:12)(cid:12)(cid:12) r = r c = r c − r i r e − r i yields the following second order equation Aδ c + Bδ c + C = 0 , (26)where A , B and C are similar to the constants defined in Eqs. (55)-(57) in Yu et al. 2017 [20].The solutions for δ c (see the curve v c in Fig. 2) δ c = − B A + √ B − AC A , (27) δ c = − B A − √ B − AC A . (28)For the slow surface sausage and kink mode due to having resonance absorption, Ω /ω si shouldbe below v ci , which means that only δ c satisfies this condition [20].Next, we turn to calculate the parameter ∆ c appeared in the dispersion relation (22). To thisaim, using Eq. (25) and ω c ( r c ) = k z v c (cid:12)(cid:12)(cid:12) r = r c we obtain∆ c ≡ (cid:20) ddr (Ω − ω c ) (cid:21) r = r c = − (cid:18) ( ω − ω f ) dω f dr + ω c dω c dr (cid:19) r = r c = − ω − ω f ( r c )) ω fe − ω fi l − ω c ( r c ) l !( (cid:0) χv sei − (cid:1) δ (cid:0) χv sei − (cid:1) − ( χ − δ ( χ − χv Aei − δ (cid:0) χv Aei − (cid:1) (29) − v si (cid:16) χv sei − (cid:17) + v Ai (cid:16) χv Aei − (cid:17) v si h δ (cid:16) χv sei − (cid:17) i + v Ai h δ (cid:0) χv Aei − (cid:1) i ) r = r c , where ω f = k z v z . Here, we study the dispersion relation (22) in the weak damping limit. We first rewrite thedispersion relation as D AR + iD AI = 0 , (30)where D AR and D AI are the real and imaginary parts of Eq. (22) respectively, given by (31) D AR = ρ i (Ω i − ω Ai ) − ρ e (Ω e − ω Ae ) k ri k re Q m , (32) D AI = πρ i ρ e k z k re Sign Ω ρ c | ∆ c | (cid:12)(cid:12)(cid:12) r = r c (cid:16) v sc v Ac + v sc (cid:17) (Ω i − ω Ai )(Ω e − ω Ae ) G m . Note that in Eqs. (31) and (32) we have the complex frequency ω = ω r + iγ , in which ω r and γ are oscillation frequency and the damping rate, respectively. In the limit of weak damping, i.e. γ ≪ ω r , the damping rate γ is given as [33] γ mc = − D AI ( ω r ) (cid:18) ∂D AR ∂ω (cid:12)(cid:12)(cid:12) ω r (cid:19) − . (33)9ere, we want to simplify Eq. (33), to obtain the damping rate of surface sausage modes in theweak damping limit, i.e. γ ≪ ω r . To this aim, we first calculate ∂D AR ∂ω from Eq. (31) as follows ∂D AR ∂ω = 2 ρ i Ω i − ρ e Ω e k ri k re Q m − ρ e (cid:0) Ω e − ω Ae (cid:1) (cid:18) k re dk ri dω − k ri k re dk re dω (cid:19) Q m − ρ e (cid:0) Ω e − ω Ae (cid:1) k ri k re dQ m dω . (34)Now from Eq. (16), one can obtain dk ri dw = − Ω i (Ω i − ω ci )( v si + v Ai )(Ω i − ω ci ) k ri , (35) dk re dw = − Ω e (Ω e − ω ce )( v se + v Ae )(Ω e − ω ce ) k re . (36)With the help of Eqs. (35) and (36) dQ m dω = xP m Ω i (Ω i − ω ci )( ω si − Ω i )( ω Ai − Ω i )(Ω i − ω ci ) + yS m Ω e (Ω e − ω ce )( ω se − Ω e )( ω Ae − Ω e )(Ω e − ω ce ) . (37)Replacing this into Eq. (34) yields ∂D AR ∂ω = 2 ρ i Ω i − ρ e Ω e k ri k re Q m − ρ e (cid:0) Ω e − ω Ae (cid:1) k ri k re (cid:18) ( Q m + xP m )(Ω i − ω ci )Ω i ( ω si − Ω i )( ω Ai − Ω i )(Ω i − ω ci ) − ( Q m − yS m )(Ω e − ω ce )Ω e ( ω se − Ω e )( ω Ae − Ω e )(Ω e − ω ce ) (cid:19) , (38)where P m ≡ I ′′ m ( x ) I m ( x ) − I ′ m ( x ) I m ( x ) ! K m ( y ) K ′ m ( y ) ,S m ≡ (cid:18) − K ′′ m ( y ) K m ( y ) K ′ m ( y ) (cid:19) I ′ m ( x ) I m ( x ) , (39)and x = k ri r i and y = k re r e . Finally, substituting Eqs. (32) and (38) into Eq. (33) one can getthe damping rate γ in the limit of weak damping for the surface modes in the slow continuumas γ mc (cid:12)(cid:12)(cid:12) ω = ω r = − πρ e k z k re ρ c Sign Ω | ∆ c | (cid:12)(cid:12)(cid:12) r = r c (cid:16) v s v A + v s (cid:17) (Ω i − ω Ai )(Ω e − ω Ae ) G m (cid:16) Ω i − χ Ω e k ri k re Q m (cid:17) − χT m , (40)where T m = (cid:0) Ω e − ω Ae (cid:1) k ri k re (cid:18) ( Q m + xP m )(Ω i − ω ci )Ω i ( ω si − Ω i )( ω Ai − Ω i )(Ω i − ω ci ) − ( Q m − yS m )(Ω e − ω ce )Ω e ( ω se − Ω e )( ω Ae − Ω e )(Ω e − ω ce ) (cid:19) . (41)Equation (40) can be more simplified in the long wavelength limit which we do in the nextsubsection. 10 .2 Weak damping rate in long wavelength limit - slow continuum In the limit k z R ≪ k ri R ( k re R ) ≪ γ , by using the asymptotic expansion of Q m , G m , P m and S m . For the sausage( m = 0) mode in the slow continuum we obtain (see Appendix A) γ c = 2 πχ SignΩ | ∆ c | R " ω ci ω si (cid:0) Ω e − ω Ae (cid:1) ω Ai ω ci + 8 χω Ai ω si (cid:0) Ω e − ω Ae (cid:1) ln( k z R ) ( k z R ) ln ( k z R ) . (42)For the kink ( m = 1) mode in the slow continuum we obtain (see Appendix B) γ c = − πχ Sign Ω8 | ∆ c | R ω ci (cid:0) Ω e − ω Ae (cid:1) ω Ai (cid:0) ω ci ω Ai − χω si (cid:0) Ω e − ω Ae (cid:1)(cid:1) ( k z R ) . (43)Under magnetic pore condition ( v Ae = 0) γ c = 2 πχ Sign Ω | ∆ c | R (cid:20) ω ci ω si Ω e ω Ai ω ci + 8 χω Ai ω si Ω e ln( k z R ) (cid:21) ( k z R ) ln ( k z R ) , (44) γ c = − πχ Sign Ω8 | ∆ c | R ω ci Ω e ω Ai (cid:0) ω ci ω Ai − χω si Ω e (cid:1) ( k z R ) . (45)In the absence of flow ( v zi = v ze = 0), Ω e = ω ci so γ c = 2 πχ Sign Ω | ∆ c | R (cid:20) ω ci ω si ω Ai + 8 χω Ai ω si ln( k z R ) (cid:21) ( k z R ) ln ( k z R ) , (46) γ c = − πχ Sign Ω8 | ∆ c | R ω ci ω Ai (cid:0) ω Ai − χω si (cid:1) ( k z R ) , (47)where these relations are the same Eqs. (79) in [22] and (38) in [20] respectively. In this section we solve the dispersion relation (Eq. (22)) numerically to obtain the frequenciesand damping rates of the slow surface sausage and kink modes and we compare the analyticalresults (Eq. 40) with the numerical results. Under the magnetic pore conditions, following[23] we set again the model parameters as v Ai = 12 km s − , v Ae = 0 km s − (i.e. B ze = 0), v si = 7 km s − , v se = 11 . − , v ci = 6 . − ( ≃ . v si ) and v ce = 0 km s − .We have assumed the flow outside the tube to be zero ( v ze = 0 km s − ). Note that the disper-sion relations, Eqs. (20) and (22), are symmetric under the exchange ( ω, v z ) with ( − ω, − v z ).Therefore, it is sufficient to consider only the positive values of flow velocity with both positiveand negative values of oscillation frequency, i.e. forward and backward waves in the presence ofupward plasma flow. Our numerical results are shown in Figs. 3 to 10.Figures 3 and 4 represent variations of the phase speed (or normalized frequency) v/v si ≡ ω r /ω si ,Doppler shifted phase speed Ω /ω si and the damping rate − γ c /ω r ( γ c /ω r ) of the slow surfacesausage modes for forward and backward waves versus k z R for various flow parameters and var-ious thickness of the inhomogeneous layer l/R = (0 . , . v zi /v si = (10 − , . , . , . , .
8) (i) The valueof the phase speed v/v si increases with increasing the flow parameter v zi /v si . (ii) The minimumvalue of the Doppler shifted phase speed decreases with increasing the flow. (iii) The maximumvalue of − γ c /ω r increases, and for low flow parameter correspond to smaller k z R when v zi /v si increases but for high flow parameter correspond to larger k z R when v zi /v si increases. (iv) Thedashed-line curves in these figures represent the analytical results of the damping rate − γ c /ω r evaluated by Eq. (40). These curves show that for the weak damping (i.e. γ c ≪ ω r ) and inthe long wavelength limit (i.e. k z R ≪
1) the oscillation frequency is not affected by the pres-ence of the transitional layer. This is also confirmed by our numerical results. (vi) For a given l/R , the minimum value of the damping time to period ratio τ D /T = 2 π/ | γ c | decreases withincreasing v zi /v si . For instance, for the case where l/R = 0 . k z R = 1, the value of τ D /T for v zi /v si = 0 . ∼
95% less than the case where there is no flow. So, the relationbetween the damping rate (time) and the flow is of interest. Several researcher obtained similarresults for the sausage modes in photospheric conditions. Yu et al. showed that for l/R = 0 . τ D /T = 14 .
11 [20] and [22] showedthat for l/R = 0 . τ D /T = 10 . B φi /B zi = 0 .
3, while our results show that the minimum value of the dampingtime to period ratio forhigh upflow is much lower. vii) For k z R →
0, we see that the dampingrate go to zero for finite values of the flow parameter, and it is an agreement with analyticalrelation Eq. (44).The right panels in Figs. 3 and 4 we plot the phase speed (or normalized frequency) v/v si ≡ ω r /ω si , normalized Doppler Shifted Ω /ω si and the damping rate γ c /ω r of the slow surfacesausage modes for backward wave versus k z R for various flow parameters v zi /v si = (10 − , . , . , . l/R = (0 . , . γ c /ω r decreases, and it corresponds to smaller k z R when v zi /v si increases. (vi) For a given l/R , theminimum value of τ D /T increases with increasing v zi /v si . For instance, for the case where l/R = 0 . k z R = 1, the value of τ D /T for v zi /v si = 0 . ∼ v/v si ≡ ω r /ω si ,phase Doppler Shifted Ω /ω si and the damping rate − γ c /ω r ( γ c /ω r ) of the slow surface sausagemodes for forward and backward wave versus the inhomogeneous layer ( l/R ) for various flowparameters and k z R = (0 . , v zi /v si = (10 − , . , . , . , .
8) (i) the frequency increases with in-creasing the flow ( v zi /v si ). (ii) With increasing l/R for k z R ≪
1, the Doppler shifted frequencyincreases, but for k z R ≫ v ci /v si . (iii) For k z R ≪
1, the Doppler shifted frequency decrease when the flow increases, for k z R ≫
1, when the Doppler shifted frequency reaches above v ci /v si , it decreases with increasingflow and tends to the value of v ci /v si . (iv) For a given k z R , the damping rate values increasesand the damping time to period ratio values decreases with increasing flow. For example, for k z R = 2 the minimum value of τ D /T for v zi /v si = 0 . ∼
93% with respect to the casewhere there is no flow.The right panels of figures 5 and 6 show the variations of the phase speed (or normalized fre-quency) v/v si ≡ ω r /ω si , Doppler shifted phase speed Ω /ω si and the damping rate γ c /ω r ofthe slow surface sausage modes for backward wave versus the inhomogeneous layer ( l/R ) for12arious flow parameters v zi /v si = (10 − , . , . , .
3) and k z R = (0 . , l/R for k z R ≪
1, the Doppler shifted frequency decreases, but for k z R ≫ − v ci /v si . (iii) For k z R ≪
1, the magnitudeof the Doppler shifted frequency increase when the flow increases. For k z R ≫
1, the Dopplershifted frequency reaches − v ci /v si , it decreases with increasing flow and tends to the value of − v ci /v si . (iv) For a given k z R , the values of the damping rate decrease and the values of thedamping time to period ratio increase with increasing flow. For example, for k z R = 2 the mini-mum value of τ D /T for v zi /v si = 0 . ∼ − γ c /ω r increases, and forlow flow parameter correspond to smaller k z R when v zi /v si increases but for high flow parametercorrespond to larger k z R when v zi /v si increases. For a given l/R , the minimum value of τ D /T increases with increasing v zi /v si . For instance, for the case where l/R = 0 . τ D /T for v zi /v si = 0 . ∼
57% less than the case where there is no flow. Yu et al.[20] showed that for l/R = 0 .
2, a minimum value of τ D /T is about 18.8 but our result gives valueabout 2.8. It is now that Soler et al. [46] have obtained this number about 1000 for l/R = 0 . γ c /ω r increases, and itsposition moves to smaller k z R when v zi /v si increases.Figures 9-10 are similar to Figs. 5-6 but for kink modes. The results show that the effect of flowon the slow resonance absorption of sausage and kink modes is almost the same. The effect ofthe slow resonance in the presence of flow on the wave damping is significant under photosphericconditions.It should be noted that for the case there is no flow, the results are similar to the results of [20].When the flow is very small (i.e v zi /v si = 10 − ) the results overlap with the no flow case.Figure 11 shows the minimum value of damping time to period ratio ( τ D /T ) for the forwardwave of the slow surface sausage (solid line) and kink (dashed line) modes versus upflow veloc-ity ( v zi /v si ). This figure shows that when the upflow velocity increases, the minimum value ofdamping time to period ratio can be considerably reduced. For instance, for the upflow velocityvalue v zi /v si = 0 .
87, the damping time to period ratio of the surface sausage mode will reachabout 0.30. This confirms that the resonant absorption in the presence of flow can be consideredas an effective mechanism to justify the rapid damping of slow surface sausage mode observedby [23]. Note that for all the results indicated in Fig. 11, the longitudinal wave number isin the observational range i.e. k z R
5. In the observational range, the minimum number ofoscillations increases slightly for large values of v zi /v si . In this paper we studied the effect of the flow parameter on the frequencies, the damping rates inslow continuum of slow sausage and kink waves in magnetic flux tubes under solar photospheric(or magnetic pore) conditions. We considered a straight cylindrical flux tube with tree regioninside, annulus and outside in which the linear density, squared magnetic field (linear pressure)and linear flow profiles are considered in the annulus region or transitional layer. In addition,we numerically solved the dispersion relation and obtained the phase speed (or normalized fre-quency) v/v si ≡ ω r /ω si , the normalized Doppler shifted frequency, the damping rate γ mc /ω r ,and the damping time to period ratio τ D /T of the slow surface sausage and kink modes forforward and backward waves under photospheric (magnetic pore) conditions. Our results showthat: 13 For forward waves, the frequency and the damping rate increase when the flow parameterincreases but for backward waves, the frequencies and the damping rate decreases whenthe flow parameter increases. • For forward waves, the damping time to period ratio decreases when the flow parameterincrease but for backward waves, the damping time to period ratio increase when the flowparameter increases. • For a given l/R , the Doppler shifted frequency, approach Ω /ω si → v ci /v si for forwardwaves and approach Ω /ω si → − v ci /v si for backward waves and γ mc /ω r → • For a given k z R , the maximum value of γ mc /ω r (or minimum value of τ D /T ) increases (ordecreases) for forward waves and decreases (increases) for backward waves. • For the case where l/R = 0 .
1, the minimum value of τ D /T for v zi /v si = 0 .
6, for instance,changes ∼
89% less for forward sausage waves and for backward sausage waves the mini-mum value of τ D /T for v zi /v si = 0 .
3, changes ∼ ∼
83% less for forward waves and ∼ • For the case of l/R = 0 . v zi /v si = 0 .
87, the damping time to period ratio of thesurface sausage mode can reach τ D /T = 0 .
30. For comparison, for a static tube (no flow)with l/R = 0 .
1, [20] obtained τ D /T = 14 .
11. This confirms that the resonant absorptionin the presence of plasma flow can justify the extremely rapid damping of the slow surfacesausage mode observed by [23].
Appendices
A Weak damping rate in long wavelength limit for the sausagemode
For the sausage mode m = 0, we have (48) Q = I ′ ( x ) K ( y ) I ( x ) K ′ ( y )= − I ( x ) K ( y ) I ( x ) K ( y ) ≈ xy (ln( y/
2) + γ e )2 , G = K ( y ) K ′ ( y )= K ( y ) − K ( y ) ≈ − ln( y/ − γ e − /y = y (ln( y/
2) + γ e ) , (50) P = I ′′ ( x ) I ( x ) − I ′ ( x ) I ( x ) ! K ( y ) K ′ ( y ) ≈ y (cid:18) − y (cid:19) (ln( y/
2) + γ e ) , (51) S = − K ′′ ( y ) K ( y ) K ′ ( y ) ! I ′ ( x ) I ( x ) ≈ x y/
2) + γ e ) . Inserting Equations (48)-(51) into Equation (41) yields T = (cid:0) Ω e − ω Ae (cid:1) k ri k re xy ln( y ) (cid:16) − y (cid:17) (Ω i − ω ci )Ω i ( ω si − Ω i )( ω Ai − Ω i )(Ω i − ω ci )+ xy (Ω e − ω ce )Ω e ω se − Ω e )( ω Ae − Ω e )(Ω e − ω ce ) ! , (52)where ln( y/
2) + γ e = ln( y ). In the limit k z R << i ≈ ω ci ) above relation becomes singular.To avoid singularity, we need to evaluate the quantity α . To this aim, following [20] we firstreplace Ω i = ω ci − α into Eq. (16) and get k ri ≃ k z α (cid:0) ω ci − ω si (cid:1) (cid:0) ω ci − ω Ai (cid:1)(cid:0) ω Ai + ω si (cid:1) = k z α ω ci ω si ω Ai , (53)where we have used the definition ω c ≡ ω s ω A ω s + ω A in obtaining the second equality of the aboverelation. In the next, the dispersion relation (20) in long wavelength limit ( k z R ≪
1) reads ρ i (cid:0) Ω i − ω Ai (cid:1) − k ri k re ρ e (cid:0) Ω e − ω Ae (cid:1) xy ln( y )2 = 0 . (54)Now, replacing k ri from Eq. (53) into (54), the quantity α can be obtained as follows α = χ ω ci ω Ai (cid:0) Ω e − ω Ae (cid:1) k z R ln( k z R ) , (55)where Ω e = ω ci + k z ( v zi − v ze ) and k ri = − ω Ai ω ci χω si (cid:0) Ω e − ω Ae (cid:1) R ln( k z R ) , (56)15 = x ln( y ) ω ci (cid:0) Ω e − ω Ae (cid:1) ( ω si − ω ci )( ω Ai − ω ci ) α − x ln( y ) ω ci (cid:0) Ω e − ω Ae (cid:1) ω si − ω ci )( ω Ai − ω ci ) α + x (Ω e − ω ce )Ω e ω se − Ω e )( ω Ae − Ω e )(Ω e − ω ce ) ! . (57)Now, replacing Eqs. (55) and (56) into Eq. (57) we obtain T = − ω Ai χ ω ci ω si (cid:0) Ω e − ω Ae (cid:1) k z R − ω Ai ω ci ln( y )2 χ ω si (cid:0) Ω e − ω Ae (cid:1) k z R ln ( k z R ) − ω ci ω Ai (Ω e − ω ce )Ω e χω si ( ω se − Ω e ) (cid:0) Ω e − ω Ae (cid:1) ln( k z R ) ! , (58)and Finally we reach T = − ω Ai ω ci + 8 χω si ω Ai (cid:0) Ω e − ω Ae (cid:1) ln( k z R )2 χ ω ci ω si (cid:0) Ω e − ω Ae (cid:1) k z R ln ( k z R ) , (59)substituting Eq. (59) in (40) we have γ c = − πρ e k z k re Sign Ω ρ c | ∆ c | (cid:12)(cid:12)(cid:12) r = r c (cid:16) v s v A + v s (cid:17) ( ω ci − ω Ai )(Ω e − ω Ae ) k z Rχ ω Ai ω ci +8 χω si ω Ai ( Ω e − ω Ae ) ln( k z R )2 χ ω ci ω si ( Ω e − ω Ae ) k z R ln ( k z R ) , (60)and after some algebra we get γ c = 2 πχ Sign Ω | ∆ c | R " ω ci ω si (cid:0) Ω e − ω Ae (cid:1) ω Ai ω ci + 8 χω Ai ω si (cid:0) Ω e − ω Ae (cid:1) ln( k z R ) ( k z R ) ln ( k z R ) . (61) B Weak damping rate in long wavelength limit for the kinkmode
For the kink mode m = 1, we have (62) Q = I ′ ( x ) K ( y ) I ( x ) K ′ ( y )= − K ( y ) ( I ( x ) + I ( x )) I ( x ) ( K ( y ) + K ( y )) ≈ − (cid:16) yx + xy (cid:17) , G = K ( y ) K ′ ( y )= − K ( y ) K ( y ) + K ( y ) ≈ y − + ( ln ( y/
2) + γ e ) − y + + ( ln ( y/
2) + γ e )= − y, (64) P = I ′′ ( x ) I ( x ) − I ′ ( x ) I ( x ) ! K ( y ) K ′ ( y ) ≈ − y (cid:18) − x (cid:19) , (65) S = − K ′′ ( y ) K ( y ) K ′ ( y ) ! I ′ ( x ) I ( x ) ≈ − y )) y x . Inserting Eqs. (62)-(65) into (41) yields T = (cid:0) Ω e − ω Ae (cid:1) k ri k re (cid:0) − (cid:0) yx + xy (cid:1) − xy (cid:0) − x (cid:1)(cid:1) (Ω i − ω ci )Ω i ( ω si − Ω i )( ω Ai − Ω i )(Ω i − ω ci ) − (cid:16) − (cid:0) yx + xy (cid:1) + y y )) y x (cid:17) (Ω e − ω ce )Ω e ( ω se − Ω e )( ω Ae − Ω e )(Ω e − ω ce ) ! . (66)For Ω i = ω ci − α , we obtain T = (cid:0) Ω e − ω Ae (cid:1) − x ω ci ω si − Ω i )( ω Ai − Ω i ) α − (cid:16) − x + (1 + 3 ln( y )) y (cid:17) (Ω e − ω ce )Ω e ( ω se − Ω e )( ω Ae − Ω e )(Ω e − ω ce ) ! . (67)In the next, the dispersion relation (20) in long wavelength limit ( k z R ≪
1) for m = 1 reads ρ i (cid:0) ω ci − ω Ai (cid:1) + ρ e (cid:0) Ω e − ω Ae (cid:1) (cid:20) x (cid:21) = 0 , (68)now, replacing k ri from Eq. (53) into (54), the quantity α can be obtained as follows α = χ ω ci (cid:0) Ω e − ω Ae (cid:1) ω ci ω Ai − χω si ω Ai (cid:0) Ω e − ω Ae (cid:1) k z R , (69) k ri = 4 χ ω ci ω Ai − χω si (cid:0) Ω e − ω Ae (cid:1) ω si (cid:0) Ω e − ω Ae (cid:1) R , (70)17utting Eqs. (69) and (70) in (67) and keeping only the sentence proportional to the sentence k z R we obtain T = 8 ω si ω Ai (cid:0) Ω e − ω Ae (cid:1) ω ci k z R ω ci ω Ai χω si (cid:0) Ω e − ω Ae (cid:1) − ! . (71)In the following with the help of Eq. 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L., & Goossens, M. 2009, ApJL, 695, L16620 k z R v / v s i Forward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (a) k z R -0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55 v / v s i Backward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (b) k z R / s i Forward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (c) k z R -0.868-v ci -0.86-0.855-0.853 / s i Backward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (d) k z R - / r Forward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (e) k z R / r Backward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (f) Figure 3: The left panels are for the forward sausage waves and the diagrams in (a), (b) and(c) represent the phase speed v/v si ≡ ω r /ω si , the Doppler Shifted phase speed Ω /ω si and thedamping rate − γ c /ω r as functions of k z R for various values of plasma flow. The right panelsare the same as the left panels for the backward sausage waves. For the damping rate thedashed curves represent the analytical solutions determined from Eq. (40). The dashed curvesin the other diagrams show the results obtained in the case of no boundary layer i.e. Eq. (20).Other parameters of the tube are l/R = 0 . v Ai = 12 km s − , v Ae = 0 km s − (i.e. B ze = 0), v ze = 0 km s − , v si = 7 km s − , v se = 11 . − , v ci = 6 . − ( ≃ . v si ) and v ce = 0 km s − . 21 k z R v / v s i Forward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (a) k z R -0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55 v / v s i Backward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (b) k z R ci /v si / s i Forward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (c) k z R -0.866-v ci-0.86-0.855 / s i Backward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (d) k z R - / r Forward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (e) k z R / r Backward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (f) Figure 4: Same as Fig. 3 , but for l/R = 0 . l/R v / v s i Forward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =1.4v zi /v si =0.6v zi /v si =0.8 (a) l/R -0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55 v / v s i Backward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (b) l/R ci /v si / s i Forward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =1.4v zi /v si =0.6v zi /v si =0.8 (c) l/R -0.864-v ci -0.863-0.8625-0.862-0.8615-0.861-0.8605-0.86 / s i Backward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (d) l/R - / r Forward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =1.4v zi /v si =0.6v zi /v si =0.8 (e) l/R / r -3 Backward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (f) Figure 5: The left panels are for the forward sausage waves and the diagrams in (a), (b) and(c) represent the phase speed v/v si ≡ ω r /ω si , the Doppler Shifted phase speed Ω /ω si and thedamping rate − γ c /ω r as functions of l/R for various values of plasma flow. The right panelsare the same as the left panels for the backward sausage waves. For all panels we have assumed k z R = 0 .
5, other parameters are the same as Fig. 3.23 l/R v / v s i Forward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (a) l/R -0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55 v / v s i Backward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (b) l/R ci /v si / s i Forward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =1.4v zi /v si =0.6v zi /v si =0.8 (c) l/R -0.866-v ci -0.862-0.86-0.858-0.856-0.854-0.852 / s i Backward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (d) l/R - / r Forward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (e) l/R / r -3 Backward Sausage Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (f) Figure 6: Same as Fig. 5 , but for k z R = 2.24 k z R v / v s i Forward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (a) k z R -0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55 v / v s i Backward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (b) k z R ci /v si / s i Forward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (c) k z R -0.868-0.866-v ci -0.862-0.86-0.858-0.856 / s i Backward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (d) k z R - / r Forward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (e) k z R / r Backward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (f) Figure 7: The left panels are for the forward kink waves and the diagrams in (a), (b) and(c) represent the phase speed v/v si ≡ ω r /ω si , the Doppler Shifted phase speed Ω /ω si and thedamping rate − γ c /ω r as functions of k z R for various values of plasma flow. The right panelsare the same as the left panels for the backward kink waves. For the damping rate the dashedcurves represent the analytical solutions determined from Eq. (40). The dashed curves in theother diagrams show the results obtained in the case of no boundary layer i.e. Eq. (20). For allpanels we have assumed l/R = 0 .
1, other parameters are the same as Fig. 3.25 k z R v / v s i Forward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (a) k z R -0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55 v / v s i Backward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (b) k z R ci /v si / s i Forward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (c) k z R -0.87-0.865-0.86-0.855 / s i Backward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (d) k z R - / r Forward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (e) k z R / r Backward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (f) Figure 8: Same as Fig. 7 , but for l/R = 0 . l/R v / v s i Forward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (a) l/R -0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55 v / v s i Backward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (b) l/R ci /v si / s i Forward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (c) l/R -0.864-v ci -0.8635-0.863-0.8625-0.862-0.8615 / s i Backward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (d) l/R - / r -3 Forward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (e) l/R / r -3 Backward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (f) Figure 9: The left panels are for the forward kink waves and the diagrams in (a), (b) and(c) represent the phase speed v/v si ≡ ω r /ω si , the Doppler Shifted phase speed Ω /ω si and thedamping rate − γ c /ω r as functions of l/R for various values of plasma flow. The right panelsare the same as the left panels for the backward kink waves. For all panels we have assumed k z R = 0 .
5, other parameters are the same as Fig. 3.27 l/R v / v s i Forward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (a) l/R -0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55 v / v s i Backward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (b) l/R ci /v si / s i Forward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (c) l/R -0.866-v ci -0.862-0.86-0.858-0.856-0.854-0.852 / s i Backward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (d) l/R - / r Forward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.2v zi /v si =0.4v zi /v si =0.6v zi /v si =0.8 (e) l/R / r -3 Backward Kink Waves v zi /v si =0v zi /v si =10 -5 v zi /v si =0.1v zi /v si =0.2v zi /v si =0.3 (f) Figure 10: Same as Fig. 9 , but for k z R = 2.28 v zi /v si D / T Forward Waves
SausageKink
Figure 11: The minimum value of the damping time to period ratio ( τ D /T ) for the forwardwaves including the slow surface sausage (solid line) and kink (dashed line) modes versus upflowvelocity ( v zi /v si ) for l/R = 0 .
1. Here, we have k z R ≤≤